Every meromorphic function is the Gauss map of a conformal minimal surface

Abstract

Let $M$ be an open Riemann surface. We prove that every meromorphic function on $M$ is the complex Gauss map of a conformal minimal immersion $M\to {\mathbb{R}}^3$ which may furthermore be chosen as the real part of a holomorphic null curve $M\to {\mathbb{C}}^3$. Analogous results are proved for conformal minimal immersions $M\to {\mathbb{R}}^n$ for any $n>3$. We also show that every conformal minimal immersion $M\to {\mathbb{R}}^n$ is isotopic through conformal minimal immersions $M\to {\mathbb{R}}^n$ to a flat one, and we identify the path connected components of the space of all conformal minimal immersions $M\to {\mathbb{R}}^n$ for any $n\ge 3$.

J. Geom. Anal., to appear. Available on SpringerLink: https://link.springer.com/article/10.1007%2Fs12220-017-9948-3

References in corpus (1)

• The parametric h-principle for minimal surfaces in ${\mathbb{R}}^n$ and null curves in ${\mathbb{C}}^n$

Cited by in corpus (5)

• New complex analytic methods in the theory of minimal surfaces: a survey
• Holomorphic Legendrian curves
• Interpolation by conformal minimal surfaces and directed holomorphic curves
• Interpolation by complete minimal surfaces whose Gauss map misses two points
• Minimal surfaces for undergraduates